PHYSICAL REVIEW A VOLUME 44, NUMBER 6 15 SEPTEMBER 1991

Density-functional theory of the nematic phase:Results for a system of hard ellipsoids of revolution

Jokhan Ram and Yashwant SinghDepartment ofPhysics, Banaras Hindu University, Varanasi 221 005, India

(Received 14 February 1991)

A second-order density-functional theory is used to study the isotropic-nematic transition in a systemof hard ellipsoids of revolution. The direct pair-correlation functions of the coexisting isotropic liquidthat enter in the theory as input information are obtained from solving the Ornstein-Zernike equationusing the Percus-Yevick closure relation. The spherical harmonic expansion coe%cients of the correla-tion functions obtained from this solution are in good agreement with those found from computer simu-

lations. We find that a system spontaneously transforms to a nematic phase when the structural parame-ter denoted by cz2' attains a value close to 4.40. This value of c22' depends, although very weakly, on thevalue of the length-to-width ratio of the molecules. The transition parameters we found are in very goodagreement with the results generated by computer simulations. By using the harmonic coef5cients of thedirect pair-correlation functions, we have calculated the Frank elastic constants of the nematic phase.

I. INTRODUCTK)N

Systems consisting of long elongated or disk-shapedmolecules do not usually show a single transition from acrystalline solid to an isotropic liquid or vice versa, butrather a cascade of transitions involving new phases; thesymmetry and mechanical properties of these phases areintermediate between those of liquid and crystal andtherefore have often been called liquid crystals [1—5].Over the past 15 years such phase diagrams have beenstudied by numerical simulations of orientable moleculesof various sorts [6—13]. Some of these simulations uselattice models so that they are suitable only for the orien-tationally ordered phases (i.e., for plastic solid to orderedcrystal transition). Other simulation studies, however,use continuous models with hard ellipses (in two dimen-sions), hard spherocylinders, and hard ellipsoids of revo-lutions (in three dimensions). A system of hard ellipsoidsof revolutions (HER) has revealed four distinct phases,viz. , isotropic Quid, nematic Quid, plastic solid, and or-dered solid [8]. A system of parallel spherocylinders (SC)exhibits smectic- A phase [12].

Starting with the work of Singh and Singh [14] manyworkers [1S—18] have studied isotropic-plastic solid (IP)and isotropic-nematic (IN) transitions in a system ofHER. All of these workers have essentially used asecond order densit-y functional th-eory (DFT) of freezingof Ramakrishnan and Yussouff (RY) [19]which examinesat a mean field level the energetic and entropic difFerencesbetween the uniform and ordered phases at a given tem-perature. The entropy is assumed to be the entropy asso-ciated with rearranging the cell of particles with dimen-sion of the order of the correlation length. The structuralenergy is estimated to quadratic order in a density expan-sion using the direct pair-correlation function (DPCF) ofthe coexisting fIuid. The ordered phase is regarded as acalculable perturbation on the Quid.

Compared to atomic Auids, our knowledge of thecorrelation functions of the isotropic fIIuid of nonspheri-

cal molecules is meager. Attempts have been made toguess them using thermodynamics of the Quid obtainedfrom simulations as a guide. When these correlationfunctions were used in the DFT to locate the transitions,results were found which were not in agreement with theknown results. Numerical methods have now becomeavailable to solve integral equation theories [hypernettedchain (HNC) and Percus-Yevick (PY) equations] of theisotropic fIuid. The correlation functions found fromthese theories can be used in the DFT to calculate thefreezing parameters. Singh, Mohanty, and Singh [16],have solved the PY equation for systems of HER andhard dumbbells and have studied the IP transition. Re-sults found for both systems are in good agreement withthe simulation results. Perera and co-workers [20] havesolved the HNC and PY equations for systems of HER,HSC (hard spherocylinders), and two model tluidscharacterized by pair potential of a generalized Maier-Saupe type, and have studied IN transition. A surprisingresult found by them is that the PY theory does not givean IN transition for any of the Auids they considered.Recently Talbot, Perera, and Patey [21] have donemolecular-dynamics (MD) calculations for fluid of HERand have shown that both the HNC and PY theories arein reasonable agreement with MD results for prolate mol-ecules.

In this paper we solve the PY integral equation numer-ically for systems of HER at densities which span beyondthe range studied by Perera and co-workers [20] andshow that the IN transition takes place for ellipsoids oflength-to-width ratio xo ~3.0. We compare our resultswith the computer simulation values of Frenkel, Mulder,and McTague [8] and with the results of other workers.The paper is organized as follows. In Sec. II we summa-rize the density-functional formalism applied to study thefreezing of molecular fIuids. Section III discusses thesolution of the PY equation for the isotropic Quid using amethod in which the pair-correlation functions are ex-panded in a series of suitably chosen angle-dependent

1991 The American Physical Society

DENSITY-FUNCTIONAL THEORY OF THE NEMATIC PHASE: 3719

basis functions (spherical harmonics). In Sec. IV we dis-cuss the results for the IN transition. Section V is devot-ed to the calculation of the Frank elastic constants of thenematic phase using the harmonic coe%cients of theDPCF found from solving the PY integral equation. Thepaper ends with a brief discussion given in Sec. VI.

II. SYSTEMATICSOF THK DENSITY-FUNCTIONAI. THEORY

OF FREEZING

The physical idea behind the density-functional theoryof freezing is the fact that at the first-order transition thecorrelation length remains of the order of a few molecu-lar diameters [22]. All phenomena at distances largerthan the correlation length are therefore treated in amean field approximation. We can imagine dividing theliquid up into cells with a diameter of the order of corre-lation length centered at a point with coarse grained localnumber density p(x) (also referred to as singlet distribu-tion). The vector x is taken here to indicate both the lo-cation r of the center of a molecule and its relative orien-tation Q described by the Euler angles 8, P, and g. Thesinglet distribution p(x) is defined as

p(x)—:p(r, Q) =X(5(r—r')5(Q —Q') ) (2.1)

where 5(a b) is th—e Dirac 5 function and ( ) representsensemble average over the positions r' and orientationsQ' of the N particles in the system. For a uniform Quid,p(r, Q) is a constant independent of positions and orienta-tions. We use the notation dQ=(1/Q)(sin8)d8dgdg.Note that Q is equal to 4m for linear molecules and 8mfor molecules of arbitrary shapes. For the ordered phase,the singlet distribution is expressed in terms of Fourierseries and the Wigner rotation matrices. Thus

the singlet distribution [23],

1p(x) =—exp[Pp+c"'(x)]

A(2.4)

where P= ( kii T )' is the inverse temperature, p is the

chemical potential, and A the cube of the thermal wave-length associated with a molecule. —kTc"'(x) is the sol-vent mediated potential field at x. The function c'"(x) isrelated to the Ornstein-Zernike DPCF c' '(x„x2)by therelation

5c"'(x, ) =C'"(X„X,)5p x2

(2.5)

The excess Helmholtz free energy

PEA =P(A —A;d) (2.6)

arising from the intermolecular interactions is also relat-ed to c"'(x) by the relation [22]

5(pb A ) (i)( (2.7)5p(x)

In Eq. (2.6), PA;d is the ideal gas part of the reducedHelmholtz free energy,

P 3;d = f dx p(x) I in[p(x)A] —1 j .

For a nonuniform system all the quantities defined aboveare functionals of p(x) which we, whenever essential, in-dicate by square brackets in the argument of the quantity.

Equations (2.4)—(2.8) are the starting equations of theDFT and have been used to develop a variety of approxi-mate forms of the free energy functionals [22]. The func-tional integration of Eq. (2.5) from some initial densitypo(x) to the final density p(x) gives

p(r, Q)=pop y g, „(G)exp(iar)D'„(Q)G l, m, n

(2.2) c"'(x;[p])—c'"(x;[p, ])

dx2bp(x2) ds c' '(x„x~;[p(x,s)]) (2.9)

Qi „(G)=2I +1 drdQp(r, Q)exp( iG r)D *„(Q—) .

(2.3)

where

bp(x) =p(x) —po(x) (2.10a)

Here Q& „(G)are the order parameters which measurethe nature and strength of the ordering and p0 is themean number density of the system. [G [ are the recipro-cal lattice vectors. The Wigner rotation matricesD „(Q)satisfy the following orthogonality condition:

f d Q D' *„.(Q)D' „(Q)= 5g 5 .5„„ll mm' nn

In the absence of external field, one formally writes, for

p(x;s)=po(x)+st(x) . (2.10b)

The parameter s in Eq. (2.9) characterizes a path of in-tegration in the space of density function. The existenceof the function PA [p] guarantees that the result of Eq.(2.9) is independent of the path of integration. Substitut-ing Eq. (2.9) into Eq. (2.7) and doing the functional in-tegration along the same path in the density space as be-fore, we get

Pb. & [p]—PEA [po] = —fdx, bp(x, )f ds c"'(x, [sp„])1 (2)dx, f dx2bp(x, )bp(x~) s ds ds'c' '(x„x2;[sp(x,s')]) .

0 0

The initial density po(x) is still undefined. If we take po(x) equal to zero, Eq. (2.11) reduces to

(2.11)

3720 JOKHAN RAM AND YASHWANT SINGH

(2.13)

phd [p]=—f dx, f dx2p(x, )p(x~) f ds f ds'c' '(x„x~;[s'p]). (2.12)

Note that Eqs. (2.11) and (2.12) are exact but need the value of c' '. Assuming that one can calculate the function c' '

for any density along the path of integration, Eq. (2.11) or (2.12) provides a useful way of calculating Pb, A [p] for anonuniform system. But it is only for a uniform Quid that c' ' is obtained either by solving the integral equationtheories of the liquid state or by adopting some approximate schemes like perturbation etc. or by computer simulations.We therefore write Eqs. (2.11) and (2.12) in terms of correlation functions of a uniform fiuid. This is done in two ways.In one of the methods we choose po(x) =p&, the density of the coexisting liquid that has chemical potential equal to thatof the ordered phase, and perform functional Taylor expansion in the ascending powers of bp(x) =p(x) —p&. The ex-pansion coeKcients are the direct correlation functions c'"' of the coexisting Quid. Thus

phd [p]—phd(p~)= —fdx(Ap(xi)c'' (p&) ——,' f dx& f dx2bp(x()c (x»x2', p&)bp(x2)

——,' fdx, fdx2bp(xi)bp(x2)c' '(x»x2, x3,'p&)bp(x3)+

Note that for a uniform fiuid c"'(p&) is a constant in-dependent of coordinates. 2 priori, we have no knowledgeof the convergence of this series. The expansion is usefulonly when the first few terms of Eq. (2.13) are important.Since in the case of molecular Quids our knowledge of c'"'for n ~ 3 is almost nil, Eq. (2.11) is useful only when allterms, except the first which involves c' ', are negligible.A theory based on this approximation is known assecond-order DFT of freezing. For atomic Quids thesecond-order DFT has been found inadequate in describ-ing the freezing and elastic constants of the solid [22].This is because the Quid-solid transition is strongly firstorder with large discontinuities in the entropy, density,and order parameters. The structure of the crystallinephase cannot be adequately expressed by the low-orderexpansion term in Eq. (2.13). The contributions of thehigher-order terms, particularly those involving c' ' andc' ', are believed to be sizable and therefore cannot beneglected.

In an alternative approach, attempts have been madeto find approximate but nonperturbative free energyfunctionals which include in some sense the contributionof all higher-order terms. For atomic systems a numberof schemes have recently been suggested to achieve thisgoal. For example, in a scheme recently suggested byLutsko and Baus [24] one defines an effective density,

p[p]= —fdrp(r) f dr'p(r')u)(~r —r'~;[p])1

(2.14)

where the weight factor m is found from the relation

f ds f ds'c( '(r;pfs'p])u)(r; [p])= f dr f ds f ds'c' )(r;s'p[p])

(2.15)

pfp] is viewed here as a functional of p(r). The excessfree energy of the ordered phase is found from the rela-tion

phd [p]= —fdrp(r) fdr'p(r') f ds f ds'c' )(~r —r';p[s'p]) . (2.16)

Note that Eq. (2.16) differs from the exact expression[given by Eq. (2.12)] of Pb, A in that the density averagedDPCF of the solid has been replaced by the DPCF on aneffective liquid of density p. Since Eq. (2.16) has the samefunctional dependence on p(r) as the exact Pb, A [p], itfollows that the functional relation between 6A and theDPCF, W=pA —

p)M f dxp(x) . (2.18)

I

DFT in this article to locate the IN transition and to cal-culate the Frank elastic constants of the nematic phasenear the transition point.

The grand thermodynamic potential which is generallyused to locate the transition is defined as

(„) 5"Pb, A [p]

II ~p(r)' (2.17) Substituting the value of PA from Eqs. (2.8) and (2.13)into Eq. (2.18) we get, correct to second order in bp(x),

is preserved by Eq. (2.16). Though the method describedhere can in principle be extended in a straightforwardway to molecular Quids, the numerical calculations arequite complicated because of the presence of a large num-ber of variables. Moreover we find that (see Sec. VI) forthe case of IN transition, the effective density p is close tothe density of the coexisting isotropic liquid. This indi-cates that the contribution of higher-order terms in Eq.(2.13) is negligible. We therefore use the second-order

58 = 8 —8'~=68'j +68 2 (2.19)

and

where 8'& is the grand thermodynamic potential of theisotropic liquid and

1 fdrdQjp(r, Q)ln[p(r, Q)/p/] —bp(r, Q)]p~V

(2.20)

DENSITY-FUNCTIONAL THEORY OF THE NEMATIC PHASE: 3721

58'2

1V

1J «»dQ&dQ&&p(ri, Qi)

2pf V

where b p' =(p„—pf ) /pf is the relative change in thedensity at the transition and

Xc' '(r, ~, Q„Q~)hp(r~,Q~) . (2.21) PI =—,' f f ( Q)P&(cos8)sin8d 8 (2.26)

The ordered phase density is found by minimizing 6 Wwith respect to arbitrary variation in the ordered phasedensity subject to the constraint that there is one mole-cule per lattice site (for perfect crystal) and/or orienta-tional distribution is normalized to unity (see below).Thus

~i + dripdQpc (rip Qi Qp pf )p(r, Q) (2)

pfX b.p(r~, Q~) (2.22)

p(r, Q) =p„f(Q) (2.23)

where A,L is a Lagrange multiplier which appears in theequation because of constraint imposed on the minimiza-tion.

In order to locate the freezing transition one attemptsto find the solution of p(x) of Eq. (2.22) which has sym-metry of the ordered phase. Below a certain liquid densi-ty, say p', the only solution of Eq. (2.22) is the uniformliquid solution p(x) =pf. Above p' a new solution of p(x)can be found which corresponds to the ordered phase.The phase with lowest grand potential is taken as thestable phase. The transition point is determined by thecondition 68'=0. Implicit in this approach is an as-sumption according to which system is either entirelyliquid or entirely ordered phase; no phase coexistence ispermitted. This signifies the mean field character of thetheory.

An important ingredient of the DFT is the actual formassumed for the density in parameter space. If the fullsearch for the parameter space were performed the exactansatz for the density would be irrelevant. Since we haveto truncate when performing the computation, it is help-ful to choose an efficient ansatz. One such form of p(x) isgiven by Eq. (2.2). Since we are concerned here with thefreezing of fluids consisting of cylindrically symmetricmolecules to a uniaxial nematic phase we rewrite Eq. (2.2)as

is the orientational order parameter of the nematic phase.The prime on the summation in Eq. (2.25) indicates thecondition that I is even. p„is the number density of thenematic phase. The orientational singlet distributionf(Q) is normalized to unity, i.e.,

fdQf(Q)=1. (2.27)

For a uniaxial phase of cylindrically symmetric moleculesf(Q) depends only on angle 8 between the director andthe molecular symmetry axis.

It has often been found convenient in the case of thenematic phase to use the following ansatz for f(Q):

f(Q)= Aoexp g A&P&(cos8)1

(2.28)

58 I /N= —Ap*+pf

P„A0ln +g' A, (Pi

Pf 1~2

(2.29)

The interaction term h8'2/X is evaluated using Eq.(2.25) for f(Q). Thus

SW /N= —-'yP P c"'2 g 11 12 I I 12

ll, l~

where the structural parameter c 1 1 is given as

(2.30)

c 'I I= (2l, + 1)(2l&+ 1)pf

X Jdr d Q, d Qzc' '(r, Qi, Qz)

A0 is determined from the normalization condition ofEq. (2.27). In the A, l ~0 limit f(Q)—+1 corresponding tothe isotropic phase. For finite A, &, f(Q) is peaked about0=0 and m", these angles correspond to parallel alignmentof the molecules.

Following Singh, Mohanty, and Singh [16] we evaluatethe entropy term b, W, /N using the ansatz (2.28). Thus

with XP& (cos8, )PI (cos8z) . (2.31)

and

p„=pf(1+hp*)

f(Q ) = 1+ g' (21+ 1 )PiPi(cos8)1~2

(2.24)

(2.25)

All angles in Eq. (2.31) refer to the space-fixed Z axis.Note that c 0O'—=c0 is related to the isothermal compressi-bility.

Since the isotropic liquid DPCF is an invariant pair-wise function, it has an expansion of the form

c (r, Q»Qz)= g g c. . .(r)C (l, lzl;m, mmmm)Yi (Q, )Y, (Qz)Y,* (r)1I, l~, 1 m l, m~, m

(2.32)

~(0)c1l l~

where C~(lilzl; m, mmmm ) are the Clebsch-Gordan coefficients, ci' i' i(r ) the spherical harmonic coefficient of the DPCF1 2

in a space-fixed frame, and r =r/~ r~ is a unit vector along the intermolecular axis. From Eqs. (2.31) and (2.32) we get

(21, + 1)(21~+1)pf C ( l i i& 0;000)f dr r c~' i' o ( r ) . (2.33)

3722 JOKHAN RAM AND YASHWANT SINGH

From Eqs. (2.22) and (2.32) we find

(1+bp )5io+Pi= 1 dQiP&(cosOi)exp AL+pf g [(2li+1)(2lz+I)] Cg(lilz0;000)Pi(cosOi) Jdrr ci'iso(r)11,12

(2.34)

where A,L is a Lagrange multiplier. To solve the aboveequations we need to know ci i' i(r) In. the following sec-

1 2

tion we describe the PY theory to obtain these harmoniccoefficients.

where

and

b(1,2) =1—exp[Pu(1, 2)]

III. PAIR-CORRELATION FUNCTIONSOF THE ISOTROPIC PHASE: PY THEORY

The Ornstein-Zernike (OZ) equation

h(1, 2) =c(1,2)+pf jdx3c(1,3)h(2, 3) (3.1)

c(1,2)=b(1,2),g(1,2)=f(1,2)[g(1,2) —c(1,2)] (3.2)

couples the pair-correlation function h(1, 2)=g(1,2) —1

and the DPCF c(1,2) for an isotropic fiuid of density pf.Here we introduce two notational simplifications. Wewrite n instead of I„in the argument of a function anddrop the superscript from functions c and h because thecorrelation functions appearing here and below are onlytwo-body functions. The PY closure relations are writtenin various equivalent forms; the form which we favorhere is

f(1,2)=exp[ —Pu(1, 2)]—1 .

Here u (1,2) is a pair potential energy of interaction. Wecan expand functions appearing in Eqs. (3.1) and (3.2) inspherical harmonics either in space-fixed (SF) frame or inbody-fixed (BF) frame. The SF and BF frame sphericalharmonic coefficients are defined as Ai i i(r ) and

1 2

Ai i (r), respectively, for an arbitrary function A(1,2)1 2

and similarly their k-space analogs.In the k space, the OZ equation has a particularly sim-

ple form in terms of BF spherical harmonic coefficients:

fi i ~(k) ci i (k—)+(—1) gfi i (k)ci i (k)Pf

'3

(3.3)

where the summation is over allowed values of l3. Simi-larly from Eq. (3.2) we get, in BF frame,

Cl| l~m ( F12 )—ii 4&

17 1

12,12

(2l', + 1)(212+1)(2l", + 1)(2l2'+ 1)C (l il2'1 i', 000)C (l2l2'lq', 000)

X g Cg(l', l", I, ;m'm "m )Cg(l2l2'l2', m'm "m )bi. i. ,(&i2)gi. i- „(ii2)I tf 1 2 1 2

7

(3.4)

4mc (r )=~I1 121 12

m

1 /2

Cs (I i 121;m i m 2m )ci i ~ (r i2 )

(3.5a)

or1/2

2l+14~

C ( 1, l zl; m, m z m )

Xci i i(ri2) (3.5b)

where the summation is over allowed values of I'„l", , lz,l z', m', and m". A similar expression can be written inthe SF frame. Numerically one finds it easier to calculatethe BF harmonic coefficients than the SF harmoniccoefficients. The two harmonic coefficients are, however,related through a linear transformation (often called theClebsch-Gordan transformation),

Note that in any numerical calculation we can handleonly a finite number of the spherical harmoniccoefficients for each orientation-dependent function. Theaccuracy of the results depends on this number. As theanisotropy in the shape of molecules (or in interactions)and the value of the Quid density pf increase more har-monics are needed to get proper coverage. If we truncatethe series of Eqs. (3.3) and (3.4) at the value of l indices 2[i.e., consider only five harmonics listed in the first twocolumns of Eq. (3.6) for c harmonics] we find thestructural parameters c zz' are underestimated by about12.2% at pf d 0 =0.309. But when we include all harmon-ics up to I indices 4, we find the value of c 22' is fully con-verged and remains unchanged (less than 0.5%) evenwhen all harmonics with I indices 6 are included in thesolution of the OZ equation. The only effect thesehigher-order harmonics (i.e., harmonics with l indices 6)appear to have on the lower-order harmonics is to modify

DENSITY-FUNCTIONAL THEORY OF THE NEMATIC PHASE: 3723

C 000 ( r 12 ) 200 ( 12 )

220( 12 )

C221("12 )

C222( 112 )

C400( 12 )

C420( 12 )

C 440 ( r 12 )

C421("12 )

C 441( 112 )

C422(r12 )

C442( 12 )

C443( r12 )

C444(&12) .

(3.6)

The numerical scheme adopted to solve the PY equa-tion is the same as that given in the Aow diagram of Ref.[25]. While for details we refer to this reference, we listhere the main steps involved in the procedure. (a) Firstwe make an initial guess for the values of cj j (r,2 ). (b)

1 2

Taking these values of cj j ~(r,2) we calculate cj j j(r,2)1 2 1 2

using Eq. (3.5a). (c) This result is used to obtain cj j j(k )1 2

through Hankel transform. (d) The coefficients cj j j(k)1 2

are used to obtain cj j (k) through Clebsch-Gordan1 2

the finer structure of the harmonics at small values of rwhose contributions to the structural parameters arenegligible. All the results we report in this paper are ob-tained with 14 harmonics (listed below for c function) foreach function in Eqs. (3.3) and (3.4);

transform (3.5b). (e) Now the OZ equation Eq. (3.3)] issolved for f& j ~(k). (f) By converting j j ~(k) to

Itj j j(k) through (3.5a) and taking the inverse Hankel1 2

transform we obtain hj j j(r12). (g) This result is used to1 2

obtain hj j (r12) from Eq. (3.5b). (h) With known1 2

hj j ~(r,2) we evaluate cj j ~(r,2) from Eq. (3.4). (i)1 2 1 2

With these values of cj j (r,2 ) we go to step (b) and re-1 2

peat the other steps. This is continued until a conver-gence, for which we used the criteria

lcj j (F12) cj

is achieved. We have solved the PY equation for h and charmonics for systems of HER.

The potential energy of interaction of a pair of HER'sis represented as

ao, r12 & D(Q12)u(r12, 01,Q2) —

0 ~D(Q )(3.7)

where D(Q12)[—:D(912,Q,2)] is the distance of closestapproach of two molecules with relative orientation Q,2.For D(Q12) we use the expression given by the Gaussianoverlap model of Berne and Pechukas [26],

D ( Q, 2) =D ( r, 2, Q12)

=d0 1 —y12 el) +(r12 e2) 2+(r12 el)(r12 e2)(el e2)

1 —y (e, e2)(3.8)

where e, and e2 are unit vectors along the symmetry axesof two interacting hard ellipsoids,

X0 1

2 + (3.9)

and r, 2 is a unit vector along the intermolecular axis.x0=a/b, where 2a and 2b denote lengths of major andminor axes of the ellipsoids and d0=2b. The molecularvolume and packing fraction are, respectively, given asU=(m/6)dO x0 and 2)=(m j6)pfx0 where pf =pfdO .Note that Eq. (3.8) is a close but not exact representationof the ellipsoid overlap.

We compare in Figs. 1 —4 the results found for BF hharmonics for x0=3.0 and pf =0.24 with MD [21] re-sults. In Fig. 1 the plotted value is g(r)=h000(r)/4m+1.As mentioned in the Introduction, Talbot, Perera, andPatey [21] have recently solved the HNC and PY equa-tions using a procedure which differs from that of ours.Their values of the BF h harmonics are also plotted inthese figures. We find that our values are in good agree-ment with the MD results for all the harmonics shown in

1 ~ 0g(r)

0.5

00 I

1-5I

20I

25r/do

I

30 35I

4.0

FIG. 1. Pair-distribution function g ( r ) for xo =3.0,pf*=0.24. The solid and dashed curves are, respectively, HNCand PY results of Perera and co-workers (Ref. [20]). The dash-dotted curve is present PY result and circles are MD results ofTalbot, Perera, and Patey (Ref. [21]).

3724 JOKHAN RAM AND YASHWANT SINGH

05— 0.25—

00C.

c

CD

—0-5- —0.25

—1 01.0

I

15I

2.0I

25

r/do

I

30I

35I

4.0-0-50 '

1.0I

1 5I

20I

25r/do

I

3-0I

3.5I

4.0

FIG. 2. The spherical harmonic coeKcient h20O(r)/4m inbody-fixed frame for x0=3.0, pf*=0.24. The curves are thesame as in Fig. 1.

FIG. 4. The spherical harmonic coefticient h»i(r)/4m inbody-fixed frame for x0=3.0, pf*=0.24. The curves are thesame as in Fig. 1.

these figures and di6'er from the results of Talbot, Perera,and Patey. For some harmonics we find our results arecloser to their HNC values than their PY values. In Figs.5 —8 we plot our results for SF c harmonics scaled by(4~) for xo=3.5 and g=0.456. It can be seen fromthese figures that the most important coefficients arethose for which l=O and all coefficients with l=4 aresmall. In Fig. 9 we compare the pressure versus densityderived from the DPCF compressibility sum rule to sirnu-lation [8] results for the isotropic phase, and to the HNCand PY results of Perera and co-workers [20] forx0=3.0. We find that our PY results are slightly superi-or to the PY results of Perera and co-workers.

In a decoupling approximation the DPCF for a fluid ofhard-core rnolecules is assumed to resemble the hardsphere DPCF with the sphere diameter replaced by thedistance of closest approach.

c(r,2, Q„Q2)=c(r,2/D(r, z, Q, 2) )

CHS( r12i 1) (3.10)

20. 0

where r &2 =r &2/D(r&2, Q&2). An analytical expression forcHs(r ',z, g ) has been found from the PY equation in termsof the packing fraction g. Though the decoupling ap-proximation introduces anisotropy in the pair correlationand is exact at p —+0, since at p~0,

c(r,2, Q„Q2)-exp[ —Pu (r &z, Q&, Qz) ]—1,it cannot be exact at liquid density. When Eq. (3.10) wassubstituted in the compressibility equation, one gets equa-

0.0

COCV

1 ~ 0—

0.5

0.0

rv -20. 0Pl

K4

-40 0Il5OO

-60.0

-80.0

—051.0

I

1 ~ 5I

2.0I

2.5

'/do

I

3.0I

35I

4.0 —100.00.0 1.0

l

2.0 3.0 4.0

FIG. 3. The spherical harmonic coefficient h»o(r )/4m in thebody-fixed frame for x0=3.0, pf*=0.24. The curves are thesame as in Fig. 1.

rldp

FIG. 5. The space-fixed spherical harmonic coefficientcooo(r)/(4m) at x0=3.5 and q=0.456.

DENSITY-FUNCTIONAL THEORY OF THE NEMATIC PHASE: 3725

2.0

fV

K

I

U

0.0

—5.0

-10.0

P4

IC

I

'U

0.0

-2.0

-4.0

—6.0

-e.o0 0

I

1.0

r/do

2.0 3.0 4.0

-20.00.0 1.0 2.0

r/do3.0 4.0

FIG. 8. The space-fixed spherical harmonic coefficientsc4gp( r), c~2(r), c444( r), and c44,6(r) at xp =3.5 and g =0.456.

FIG. 6. The space-fixed spherical harmonic coefficientscpp2(r ), c22p(r ), c222(r ), and c»4(r ) at xp =3.5 and &=0.456. proximation is to use the following relation:

c(r12 Ql Q2) cHs (rl 2)[1+/ Pz(ei ez)j (3.11)

tion of state for the isotropic Quid of HER which are ingood agreement with the simulation result even for largexo. But when it is used to calculate structural parameterc I I from Eqs. (2.31) and (2.32) one finds that the IN

1 2

transition is found to take place at much lower densitythan the simulation value. This is because the structuralparameters c I I for I„i&&0are highly overestimated for

large xo.One possible way to improve upon the decoupling ap-

where the parameter P is a function of g and xo and isdetermined from the condition that g(r, 2, Q„Qz)is zero

11.0

10.0

9.0

7. 0

P4

IC

t

0 ' 5

0.0

gp 6.0

5.0

4-0

3.0

-0.5 2.0

1.0

-1.00.0

0.0I

0.1I

0.2I

0.3

-1.50-0

I

1.0 2.0

t /do

I

3.0 4.0

FIG. 7. The space-fixed spherical harmonic coefficientsc242(r), c2~(r), c246(r), and c4p4(r), at xp =3.5 and g=0.456.

FIG. 9. Pressure as a function of Auid density for xp=3.0.The solid curve is the MD result of Mulder and Frenkel (Ref.[8]), the dash-dotted curve is present PY result. The open andsolid circles represent, respectively, the PY and HNC results ofPerera and co-workers (Ref. [20]). All results are obtained us-

ing the compressibility equation.

3T26 JOKHAN RAM AND YASHWANT SINGH

5.0 1.5

0. 0

—5.0

—10.0

P)K~ —15.0

C)~—20.0

-25.0

0.5

l

ll

e

S

l a

l'

/

~

t'

—30.00 ~ 0

I

1.0r /dp

2.0 3.0 4.0O. G

0.0 1.0 2.0 3.01

4.0 5.0

FIG. 10. The space-fixed spherical harmonic coefficientsc»p(r)/(4~) at xp =3.0 and q=0.485. The solid curve is thepresent result. The dotted curve marks the Marko (Ref. [18])values. The short-dashed curve denotes the decoupling resultsand the long-dashed curve is due to Gelbert and Ben-Shaul (Ref.[27]).

inside the core. Unfortunately, for this one has to solvethe OZ equation and therefore the numerical efforts in-volved are of the same order as in solving the integralequations. In another approach, Gelbert and Ben-Shaul[27] suggested to approximate the DPCF of a system ofhard-core molecules as

FIG. 12. Pair-distribution function g(r) for xp =3.0. Thedotted, dashed, and solid curves are for g =0.219 91, 37 699, and0.4855, respectively.

1c(r,2, Q„Q2)= Iexp[ Pu(r,—,2Q„Q)2]—1I .1 —g

(3.12)

We compare the harmonics c22o(r ) and c~40(r ) found bythese approximate forms of the DPCF in Figs. 10 and 11,respectively, with those found by the solution of the PYequation. It is clear from these figures that none of the

2.0 0.6

000.4

-2,0

0.2

0K

-8.0L

C3

-10.0

C) 0, 0

—0.2

—12.000

I

1.0

r /dp

2.0I

3.0 4.0-0.4

0.0 1.0 2.0r/dp

3.0 4.0 5.0

FIG. 11. The space-fixed spherical harmonic coefficientc44p(& }/(4~) . The state condition and curves are the same asin Fig. 10.

FIG. 13. The space-fixed spherical harmonic coefficient&22p(&)/(4~)' . The state condition and curves are the same asin Fig. 12.

DENSITY-FUNCTIONAL THEORY OF THE NEMATIC PHASE: 3727

amounts to the fact that the DPCF generated by the PYtheory is density independent.

IV. RESULTSFOR ISOTROPIC-NEMATIC TRANSITION

The ordered phase coexists with the isotropic liquidwhen

440 and

(Bldg, )(b, W/N) =0 (4.1)

220 b W/X=O . (4.2)

440

440

00.0 0 ~ 1 0. 2 0.3 0.4

FIG. 14. The structural parameter c 'LL at xQ=3.0. The solidand dashed curves are, respectively, HNC and PY results due toPerera and co-workers (Ref. [20]). The present results areshown by dash-dotted curves.

approximate forms of c(r,2, Q„Q2)give the correct formfor the harmonics and are therefore not reliable eventhough they predict IN transition density correctly.

In Fig. 12 we plot our values of g(r) as a function ofr*(=rldO) for xo=3.0 at q=0. 21991, 0.37699, and0.4855. Here one observes a distinct peak developing atshort range as the density is increased. This is most like-

ly due to increasing tendency of the ellipsoids to formparallel configurations. This effect is also apparent in theSF h2zo(r) harmonic coefficient plotted in Fig. 13. Wecompare the structural parameters c 22' and c 44' calculat-ed here with those found by Perera and co-workers fromtheir solutions of HNC and PY equations in Fig. 14. It isseen from this figure that our values of c '1~

' increase withdensity and deviate rapidly from the low density linearbehavior and increase very steeply in the vicinity of thephase transition. A similar behavior has been found byPerera and co-workers for the HNC results. But theirPY values show linear behavior and are too low to giveIN transition. This is a surprising result, since it

f(Q) = 1+ g (2l + 1)PI PI(cos8)l=2, 4

(4.3)

for f(Q). They found the transition to take place at verylow density. This is because of the decoupling approxi-mation which overestimates the angular correlation andtherefore grossly underestimates the stability of the iso-tropic phase. Marko [18],who has used for f(Q) a formsimilar to Eq. (2.28), has reproduced the result of Singhand Singh [14] when he puts P =0 in Eq. (3.11) to ap-proximate the DPCF. Note that when P )0, the angu-lar correlations are suppressed and as a consequence theIN transition density is raised. However, the values of A,zand A,4 reported by Marko appear to be unrealistic, since

The former is a stability condition, while the latter is aphase coexistence condition. The g; are variation param-eters appropriate for the phase under investigation. Forthe nematic phase, the variational parameters are p„,kz,and A,4. The coexisting Quid density, pf, is determined bythe condition (4.2).

The results found by using the spherical harmoniccoefBcients of the DPCF generated by the PY approxima-tion (Sec. III) are listed for xo =3.0, 3.25, 3.5, and 4.0 inTable I. We find that the Quid freezes when the structur-al parameters c zz' and c 44' attain values -4.40 and 1.12,respectively. We find that these numbers vary, thoughvery weakly, with xo,' as xo is increased the value of c 22'

decreases while the value of c 44' increases. The transitiondensity at xo=3.0 is in good agreement with the MonteCarlo (MC) value of Frenkel, Mulder, and McTague [8].No computer simulation results are, however, availablefor xo) 3.0.

In Table II we summarize results for x0=3.0 reportedby different workers for the IN transition. Singh andSingh [14] have used the decoupling approximation (3.10)for the DPCF and a truncated expansion [see Eq. (2.25)],

TABLE I. Isotropic-nematic coexistence parameters for hard ellipsoid molecules using the harmonics generated from the PY ap-proximation.

XQ

3.003.253.504.00

0.4950.4680.456 70.434 96

0.3150.27500.24920.2077

0.31660.27640.25090.2099

P2

0.55410.56440.57760.6076

P4

0.22960.23890.25260.2834

0.003 950.005 220.007 890.0110

~QQQ

—177.29—139.34—110.19—76.333

wQ22

4.40734.40194.37914.3429

wQ

1.10571 ~ 14111.22911.3725

3728 JOKHAN RAM AND YASHWANT SINGH

TABLE II. Comparison of the critical packing fractions and the order parameters for hard ellipsoidswith length-to-width ratio xo =3.0.

MCPresent workRef. [14]Ref. [20]Ref. [28]Ref. [18]Ref. [17]Ref. [8]

0.5070.4950.3090.4180.4720.4930.50810.4199

0.5170.4970.3300.4360.4840.4940.51730.4375

P2

0.55410.5470.6570.5610.0170.53330.5677

p4

0.22960.1970.358

0.243

they produce P2-0. Baus and co-workers [28] have sug-gested the use of Eq. (3.10) for the DPCF of a system ofhard ellipsoids with g replaced by g and to calculate gfrom the relation

c(r*=1;rl) =c(r*=y~(y), rl)

where

(4.4)

where b,aHs(g) is the excess free energy per particle of ahard sphere and V„,and V,„,are excluded volumes ofthe hard-core molecules and hard spheres, respectively(the index HB denotes hard body). Parson [29] has foundthis free energy functional starting with the virial pres-sure equation and using the decoupling approximation

1/xo for xo) 1

xo for xo (1is assumed to give average contact distance in the Xphase. Equation (4.4) gives a packing fractiong=g(g, xo) of the reference I phase at each xo and isused in calculating the correlation in N phase at the pack-ing fraction g. In spite of the fact that Eq. (4.4) gives areasonable estimate for the IN transition densities, itlacks physical basis.

An interesting free energy functional is found whenone substitutes Eq. (3.10) into Eq. (2.16) and adds to it thecontribution of PA;4. For the nematic phase one gets

13'/X= fdQ f(Q)[ln[p„f(Q)]—1]

+ha (rl) fdQ f dQQ(Q )f(Q )

X [ V,„,(Q„Q)/V,„,]

(4.5)

for g(r, 2, Q„Q2). Lee [17] has recently used Eq. (4.5) tolocate the IN transition in a system of HER. His resultsare also listed in the table. The results of Perera and co-workers given in the table are those found by using the charmonics of the HNC integral equation. As mentionedin the Introduction, these authors did not find the INtransition when they used their c-harmonic results gen-erated from the PY theory.

V. FRANK ELASTIC CONSTANTS

where K&, E2, and K3 are elastic constants. The 6rstterm in Eq. (5.1) gives energy associated with splay, thesecond that associated with twist or torsion, and the thirdthat associated with flexion or bend. Thus the constantsK,- characterize the free energy increase associated withthe three normal modes of deformation of the orientednematic state.

The density-functional formalism has recently beenused by Singh and Singh [30] and Singh, Singh, and Ra-jesh [31] to obtain expression for elastic constants of theliquid crystals. Their expression for K; can be written as

K, = g K, (l, , l~) (5.2)

with

The Frank elastic constants are a measure of the freeenergy associated with long-wavelength distortion of thenematic state in which the director n(r) varies in space.If the local preferred direction at the point r is parallel tothe director n(r), the free energy associated with the dis-tortion is written as

bFd= ,' f dr—[K&(Vn) +K2(n. VXn) +K3(nXVXn) ]

(5.1)

Kp (1),l~)= — —p~k~T(21)+1) P( PI3 5 1 2

X 2&5b& OCs(—l&120;000)f r cI &o(r)dr

+[ 'b( OC (1~1~2;0—00—)+&6bl, C (1~1~2;011)+Q3bI qC (1(1~2;022)]f—r c( ( 2(r)dr (5.3)

DENSITY-FUNCTIONAL THEORY OF THE NEMATIC PHASE: 3729

K2 (li, l2) — 6—pf—k~T(2li+1)' Pi PIr

X bI OC (I, lzO;000) f r ci i o(r)dr — —[bi 0Cg(l, 122;000)+&6bi 2C~(l&l22;022)] f r ci I 2(r)dr

(5.4)

and

K3'(I„l~)—— —pfk~T(2l, +1)'PI PI3 5 1 2

X i&5bi 0C (l, l~O'000) f r ci ( 0(r)dr

+[br OCg(lil22;000) —&6bl &Cg(l&l22;011)]f r ci I 2(r)dr (5.5)

Here

bi 0= ——,'I2(l~+1)

212+ 1

4m

I

trarily choose it to be equal to 0.33. With these inputdata we calculated the value of X;* and found thatK& =1.51, Kz =1.41, and K3 =5.98. In view of the ap-proximations involved, we consider the agreement withthe results of Allen and Frenkel reasonable.

(l~ —1)! (212+ I)bi, = —

—,' 12(12+ 1)

2 2 I2+1 ! 4~

(5.6)

(I2 —2)! (2lz+1)bi ~= —,'(l2 —1)l~(12+ 1)(l2+2)

Using the ci i I(r ) determined in Sec. III we have eval-1 2

uated the value of K&, K2, and K3 near the IN transition.We summarize our results for K =If;crlk&T whereo. = 8ab in Table III for xo and g given in the table. Al-len and Frenkel [32] have used the MD simulations to es-timate the values of K;* of x0=3.0 at q=0. 556. Theyfind that E& =1.28, Xz =1.19, and K3 =4.37. Notethat the packing fraction g and P2=0. 7 taken by themare much higher than those we considered and, therefore,the two results cannot be compared as such.

In order to estimate the values of K;* at the packingfraction considered by Allen and Frenkel [32] we usedthe modified weighted density approximation (MWDA)of Denton and Ashcroft [33] (described in the followingsection) to estimate the density of an effective fluid andextrapolate our results for the function fdr r er i I(r ) to

1 2

estimate its value at this effective Quid density. Sincethese workers do not mention the value of P4, we arbi-

VI. DISCUSSION

From the results tabulated and graphed above it is ob-vious that the second-order DFT provides a very gooddescription of the properties of the nematic phase. Thestructure of the nematic phase near the transition can,therefore, be approximated as a calculable perturbationof the structure of the coexisting isotropic liquid. Theshort-range angular correlation which develops eitherdue to hindered rotation (in a system of hard-core mole-cules) or anisotropy in intermolecular interactions or dueto both is already present in the isotropic phase (see Figs.12 and 13). When this correlation grows to a certainfinite value (c ~2z'-4. 40) the isotropic phase becomes un-stable and the system spontaneously transforms to anematic phase which has long-range orientational order-ing.

In the weighted density-functional method the struc-ture of the ordered phase is expressed in terms of aneffective quid of lower density than the coexisting isotro-pic fluid. There are a number of schemes (all of themhave been proposed for atomic systems) to calculate theeffective density. One such scheme of Baus and co-workers is given in Sec. II. In a relatively simple schemesuggested by Denton and Ashcroft (DA) [33] one calcu-lates the effective density from the relation

TABLE III. Elastic constants K; =K;o./k& T where cr'=8ab, for systems of prolate ellipsoid of re-volutions.

Xp

3.003.253.504.00

0.4950.4700.4570.436

P2

0.5540.5640.5780.608

p4

0.2300.2390.2530.283

0.7630.7840.7900.851

0.6330.6510.6520.686

2.4622.5122.5972.907

3730 JOKHAN RAM AND YASHWANT SINGH

p= —fdxp(x) fdx'p(x')w(x, x';p) (6.1)

where the weight factor w(x, x';p) is chosen in such away that Eq. (2.17) is satisfied exactly for n =2 and isnormalized to unity, i.e.,

fdx w(x, x',P)=1 . (6.2)

w(x, x',p) =—2b,ao(P)

c ( x, x',p ) +—Pha o' (P) (6.3)

where Vis the volume of the sample, b,ao(p) is the excessfree energy per particle of an isotropic Quid of density p,and primes on dao(p) indicate derivatives with respect todensity. Using expansions (2.2) and (2.32), respectively,for p(x) and c(x,x', p) we find for the nematic phase [34]

7TXp

12tib, a o (p )(6.4)

Using order parameters given in Table I we calculate pfrom Eq. (6.4) and find that P is very close (about 4%lower) to the coexisting liquid density, for all values of xogiven in Table I. This justifies the use of the second-order

The explicit form for w found by DA can, for molecularQuid, be written as

DFT for the study of IN transition.In principle, the extension of the freezing theory of

atomic Quids to molecular Quids is straightforward. Inpractice, however, the use of 58 to determine the stabili-ty of the ordered phase stability is much more difficult be-cause one is forced to search for minima over a widespace of functions. In addition, the presently availableliquid structure information for molecular liquids is lessaccurate compared to atomic liquids. None of the ap-proximate forms proposed in the literature for the DPCFof molecular liquids are found accurate enough to studythe properties of the nematic phases. Therefore thecorrelation functions of the isotropic Quid have to befound either by solving the integral equation theories ofthe liquid state or by computer simulations.

In a real system molecules are neither rigid nor cylin-drically symmetric. The intr amolecular structureresponse is expected to play a crucial role in relative sta-bility of the ordered phase structure. A density-functional formulation based on an interaction-site model[35] may prove better to include these effects.

ACKNOWLEDGMENT

The authors wish to thank the Department of Scienceand Technology, New Delhi for financial assistance.

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